Let f(t)=(t^2+5t+8)(3t^2+2)

f'(t)=? should be a function in terms of the variable 't', not a number f'(3)=?

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Find `f'(t)` and `f'(3)` if `f(t)=(t^2+5t+8)(3t^2+2)` :

We could multiply f(t) out using the distributive property and then take the derivative, or we could use the product rule:

(a) Using the product rule `d/(dx)(fg)(x)=f'g+fg'` we get:

`f'(t)=(2t+5)(3t^2+2)+(t^2+5t+8)(6t)`

`=6t^3+15t^2+4t+10+6t^3+30t^2+48t`

`=12t^3+45t^2+52t+10`

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So `f'(t)=12t^3+45t^2+52t+10` and `f'(3)=895`

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(b) We could multiply first:

`f(t)=3t^4+15t^3+26t^2+10t+16` so

`f'(t)=12t^3+45t^2+52t+10` as above.

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